The number of mosquitoes in Brooklyn (in millions of mosquitoes) as a function of rainfall (in centimeters) is modeled by $m(x)=-x(x-4)$ What amount of rainfall results in the maximum number of mosquitoes?
The number of mosquitoes is modeled by a quadratic function, whose graph is a parabola. The maximum number is reached at the vertex. So in order to find the rainfall when that happens, we need to find the vertex's $x$ -coordinate. The vertex's $x$ -coordinate is the average of the two zeros, so let's find those first. $\begin{aligned} m(x)&=0 \\\\ -x(x-4)&=0 \\\\ \swarrow &\searrow \\\\ -x=0\text{ or }&x-4=0 \\\\ x={0}\text{ or }&x={4} \end{aligned}$ Now let's take the zeros' average: $\dfrac{({0})+({4})}{2}=\dfrac42=2$ In conclusion, the amount of rainfall that results in the maximum number of mosquitoes is $2$ centimeters.